Find the general solution to $$ x^3y''' + 2x^2y'' - xy' + y = \frac{1}{x} $$
The problem is that I've only dealt with power series with $g(t)=0$, not an actual function. I have the solution available $$ c_1x^{-1} + c_2x + c_3x\ln x + \frac{\ln x}{4x} $$ but I have no idea how to attempt to solve this equation. Please help!
Hint:
Step 1 Solving the homogeneous problem with the ansatz $y = x^\alpha$.
Step 2 Use the reduction of order to obtain a third homogeneous solution.
Step 3 Solve for a particular solution using higher-order variation of parameter.